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On a master-slave Bertrand game model

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  • Xin, Baogui
  • Chen, Tong

Abstract

A master-slave Bertrand game model is proposed for upstream and downstream monopolies owned by different parties, in which the upstream monopolist's output is used as the main factor of production by the downstream monopolist who is a small purchaser of the upstream monopolist's output. The bifurcation of the Bertrand-Nash equilibrium is analyzed with Schwarzian derivative. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents (LLEs), bifurcation, time series diagrams and phase portraits. With the modified straight-line stabilization method, chaos control is used to improve the aggregate profits of the two oligopolists. Lastly the welfare impacts of price fluctuations and chaos controls are briefly discussed.

Suggested Citation

  • Xin, Baogui & Chen, Tong, 2011. "On a master-slave Bertrand game model," Economic Modelling, Elsevier, vol. 28(4), pages 1864-1870, July.
    • Handle: RePEc:eee:ecmode:v:28:y:2011:i:4:p:1864-1870
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    as
    1. Den Haan, Wouter J., 2001. "The importance of the number of different agents in a heterogeneous asset-pricing model," Journal of Economic Dynamics and Control, Elsevier, vol. 25(5), pages 721-746, May.
    2. Chen, Liang & Chen, Guanrong, 2007. "Controlling chaos in an economic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(1), pages 349-358.
    3. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 512-524.
    4. Fredebick V. Waugh, 1945. "Does the Consumer Benefit from Price Instability? Reply," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 59(2), pages 301-303.
    5. Baron, David P, 1970. "Price Uncertainty, Utility, and Industry Equilibrium in Pure Competition," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 11(3), pages 463-480, October.
    6. Cees Diks & Cars Hommes & Valentyn Panchenko & Roy Weide, 2008. "E&F Chaos: A User Friendly Software Package for Nonlinear Economic Dynamics," Computational Economics, Springer;Society for Computational Economics, vol. 32(1), pages 221-244, September.
    7. Sandmo, Agnar, 1971. "On the Theory of the Competitive Firm under Price Uncertainty," American Economic Review, American Economic Association, vol. 61(1), pages 65-73, March.
    8. Qi, Guoyuan & Chen, Zengqiang & Yuan, Zhuzhi, 2008. "Adaptive high order differential feedback control for affine nonlinear system," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 308-315.
    9. Bester, Helmut, 1992. "Bertrand Equilibrium in a Differentiated Duopoly," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(2), pages 433-448, May.
    10. Hołyst, Janusz A & Urbanowicz, Krzysztof, 2000. "Chaos control in economical model by time-delayed feedback method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 587-598.
    11. Bischi, Gian Italo & Kopel, Michael, 2001. "Equilibrium selection in a nonlinear duopoly game with adaptive expectations," Journal of Economic Behavior & Organization, Elsevier, vol. 46(1), pages 73-100, September.
    12. Tramontana, Fabio & Gardini, Laura & Puu, Tönu, 2009. "Cournot duopoly when the competitors operate multiple production plants," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 250-265, January.
    13. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2009. "The dynamics of Bertrand model with bounded rationality," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2048-2055.
    14. Haibo Xu & Guangrui Wang & Shigang Chen, 2001. "Controlling chaos by a modified straight-line stabilization method," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 22(1), pages 65-69, July.
    15. Kimball, Miles S, 1993. "Standard Risk Aversion," Econometrica, Econometric Society, vol. 61(3), pages 589-611, May.
    16. Myers, Robert J., 2006. "On the costs of food price fluctuations in low-income countries," Food Policy, Elsevier, vol. 31(4), pages 288-301, August.
    17. Zhang, Jixiang & Da, Qingli & Wang, Yanhua, 2007. "Analysis of nonlinear duopoly game with heterogeneous players," Economic Modelling, Elsevier, vol. 24(1), pages 138-148, January.
    18. Bischi, Gian Italo & Naimzada, Ahmad K. & Sbragia, Lucia, 2007. "Oligopoly games with Local Monopolistic Approximation," Journal of Economic Behavior & Organization, Elsevier, vol. 62(3), pages 371-388, March.
    19. Agiza, H.N. & Hegazi, A.S. & Elsadany, A.A., 2002. "Complex dynamics and synchronization of a duopoly game with bounded rationality," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 133-146.
    20. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
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